Group #140865 — subgroups

Subgroups — #140865

Order 512.

430 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1332	C₂	1	4	yes	maximal
4	#8274	C₂	1	1	yes	maximal, characteristic
5	C₂×#494	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₁₆	—	2	1	no	—
7	C₂×C₄×Q₁₆	C₂²	1	2	yes	—
8	C₄×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	#1969	—	2	1	no	—
12	#1969	C₄	1	2	yes	—
13	#1956	—	2	1	no	—
14	#1845	C₄	1	1	yes	characteristic
15	#1767	—	2	1	no	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	C₂²	1	2	yes	—
18	#1374	—	2	1	no	—
19	#1332	—	2	8	no	—
20	#1277	—	2	1	no	—
21	C₂²×Q₁₆	—	2	1	no	—
22	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×C₄×Q₈	—	2	2	no	—
25	C₂²×C₁₆	—	2	2	no	—
26	#494	—	2	4	no	—
27	C₄×Q₁₆	—	2	4	no	—
28	C₄×Q₁₆	—	2	4	no	—
29	C₄×Q₁₆	—	4	4	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×#103	—	2	2	no	—
34	C₂×#103	—	2	2	no	—
35	C₂×#97	—	4	1	no	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#391	—	2	2	no	—
43	#389	—	4	1	no	—
44	#389	—	4	1	no	—
45	#384	—	2	1	no	—
46	C₄×C₄⋊C₄	—	4	1	no	—
47	C₄×C₄⋊C₄	—	4	1	no	—
48	C₄³	D₄	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	4	no	—
50	C₄×C₁₆	—	2	4	no	—
51	#336	—	4	1	no	—
52	#335	—	2	2	no	—
53	#335	D₄	1	4	yes	—
54	C₂²×Q₈	—	2	2	no	—
55	C₂×Q₁₆	—	2	4	no	—
56	C₂×Q₁₆	—	2	4	no	—
57	C₂×Q₁₆	—	2	2	no	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₂×Q₁₆	—	4	2	no	—
60	C₂×Q₁₆	—	4	2	no	—
61	C₂×Q₁₆	M₁₆	1	4	yes	—
62	C₂²×C₈	—	4	2	no	—
63	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄⋊Q₈	—	2	4	no	—
68	C₄⋊Q₈	—	4	2	no	—
69	C₄×Q₈	—	4	4	no	—
70	C₄×Q₈	—	4	4	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄⋊C₄	—	4	1	no	—
74	C₂×C₄⋊C₄	—	4	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄²	—	4	2	no	—
78	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄²	D₈	1	1	yes	characteristic
81	C₂×C₄²	D₈	1	1	yes	characteristic
82	C₂×C₄²	SD₁₆	1	1	yes	characteristic
83	C₂×C₄²	SD₁₆	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₁₆	—	2	4	no	—
86	C₂×C₁₆	—	4	2	no	—
87	C₂×C₁₆	—	4	2	no	—
88	#107	—	2	4	no	—
89	#107	—	4	2	no	—
90	C₄⋊C₈	—	2	4	no	—
91	C₄⋊C₈	—	4	2	no	—
92	#103	—	4	4	no	—
93	#103	—	4	2	no	—
94	#103	—	4	2	no	—
95	#97	—	8	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	C₂×D₄	1	2	yes	—
102	C₄×C₈	#41	1	2	yes	—
103	#95	—	4	2	no	—
104	#95	—	4	2	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂×Q₈	—	4	2	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	—	4	2	no	—
113	C₂²×C₄	#102	1	1	yes	characteristic
114	C₂²×C₄	#102	1	1	yes	characteristic
115	C₂²×C₄	#102	1	1	yes	characteristic
116	C₂²×C₄	#102	1	1	yes	characteristic
117	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
120	Q₁₆	—	2	8	no	—
121	Q₁₆	—	4	8	no	—
122	Q₁₆	—	4	4	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	8	2	no	—
132	C₂×C₈	—	8	2	no	—
133	C₂×C₈	#104	1	2	yes	—
134	C₂×C₈	C₄×D₄	1	2	yes	—
135	C₂×C₈	#124	1	1	yes	characteristic
136	C₂×C₈	#127	1	1	yes	characteristic
137	C₂×C₈	C₂×M₁₆	1	2	yes	—
138	C₂×C₈	#131	1	2	yes	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	8	2	no	—
144	C₄⋊C₄	—	8	2	no	—
145	C₄⋊C₄	—	8	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	4	1	no	—
153	C₄²	—	8	4	no	—
154	C₄²	SD₃₂	1	2	yes	—
155	C₄²	Q₃₂	1	2	yes	—
156	C₄²	#121	1	2	yes	—
157	C₄²	AGL(1,ℤ/8)	1	2	yes	—
158	C₁₆	—	4	4	no	—
159	C₂³	C₂×#102	1	1	yes	characteristic
160	Q₈	—	4	4	no	—
161	Q₈	—	4	4	no	—
162	Q₈	—	8	2	no	—
163	Q₈	—	8	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	#358	1	2	yes	—
182	C₂×C₄	#358	1	2	yes	—
183	C₂×C₄	#359	1	2	yes	—
184	C₂×C₄	#390	1	1	yes	characteristic
185	C₂×C₄	#407	1	1	yes	characteristic
186	C₂×C₄	#408	1	1	yes	characteristic
187	C₂×C₄	#418	1	1	yes	characteristic
188	C₂×C₄	C₂×#104	1	1	yes	characteristic
189	C₂×C₄	#421	1	1	yes	characteristic
190	C₂×C₄	#459	1	2	yes	—
191	C₂×C₄	#460	1	2	yes	—
192	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
193	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
194	C₂×C₄	#510	1	2	yes	—
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	2	no	—
198	C₈	—	4	1	no	—
199	C₈	—	8	4	no	—
200	C₈	#435	1	4	yes	—
201	C₂²	#1486	1	1	yes	characteristic
202	C₂²	#1492	1	1	yes	characteristic
203	C₂²	#1982	1	1	yes	characteristic
204	C₂²	C₂×#358	1	1	yes	characteristic
205	C₂²	#2141	1	1	yes	characteristic
206	C₂²	C₂×#359	1	1	yes	characteristic
207	C₂²	#2143	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	1	no	—
213	C₄	—	8	4	no	—
214	C₄	—	16	2	no	—
215	C₄	#1490	1	1	yes	characteristic
216	C₄	#1496	1	1	yes	characteristic
217	C₄	#1955	1	1	yes	characteristic
218	C₄	#1980	1	1	yes	characteristic
219	C₄	#2208	1	2	yes	—
220	C₄	#2209	1	2	yes	—
221	C₂	#9678	1	1	yes	characteristic
222	C₂	#11658	1	1	yes	characteristic
223	C₂	#11662	1	1	yes	characteristic
224	C₂	#11672	1	1	yes	characteristic
225	C₂	#11676	1	1	yes	characteristic
226	C₂	#13251	1	1	yes	characteristic
227	C₂	#13253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#494

C₂

yes

maximal, characteristic

C₄×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#1332

C₂

yes

maximal

#8274

C₂

yes

maximal, characteristic

C₂×#494

C₄

yes

characteristic

C₂×C₄×Q₁₆

—

C₂×C₄×Q₁₆

C₂²

yes

—

C₄×C₄⋊Q₈

—

C₂×C₄⋊C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

—

#1969

—

#1969

C₄

yes

—

#1956

—

#1845

C₄

yes

characteristic

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

#1374

—

#1332

—

#1277

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×C₄

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂²×C₁₆

—

#494

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#107

—

C₂×#107

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×#103

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

#336

—

#335

—

#335

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

M₁₆

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₄

yes

—

102

C₄×C₈

#41

yes

—

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

#102

yes

characteristic

114

C₂²×C₄

#102

yes

characteristic

115

C₂²×C₄

#102

yes

characteristic

116

C₂²×C₄

#102

yes

characteristic

117

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

118

C₂²×C₄

C₂×D₈

yes

characteristic

119

C₂²×C₄

C₂×SD₁₆

yes

characteristic

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

#104

yes

—

134

C₂×C₈

C₄×D₄

yes

—

135

C₂×C₈

#124

yes

characteristic

136

C₂×C₈

#127

yes

characteristic

137

C₂×C₈

C₂×M₁₆

yes

—

138

C₂×C₈

#131

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

SD₃₂

yes

—

155

C₄²

Q₃₂

yes

—

156

C₄²

#121

yes

—

157

C₄²

AGL(1,ℤ/8)

yes

—

158

C₁₆

—

159

C₂³

C₂×#102

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#358

yes

—

182

C₂×C₄

#358

yes

—

183

C₂×C₄

#359

yes

—

184

C₂×C₄

#390

yes

characteristic

185

C₂×C₄

#407

yes

characteristic

186

C₂×C₄

#408

yes

characteristic

187

C₂×C₄

#418

yes

characteristic

188

C₂×C₄

C₂×#104

yes

characteristic

189

C₂×C₄

#421

yes

characteristic

190

C₂×C₄

#459

yes

—

191

C₂×C₄

#460

yes

—

192

C₂×C₄

C₂×SD₃₂

yes

characteristic

193

C₂×C₄

C₂×Q₃₂

yes

characteristic

194

C₂×C₄

#510

yes

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

#435

yes

—

201

C₂²

#1486

yes

characteristic

202

C₂²

#1492

yes

characteristic

203

C₂²

#1982

yes

characteristic

204

C₂²

C₂×#358

yes

characteristic

205

C₂²

#2141

yes

characteristic

206

C₂²

C₂×#359

yes

characteristic

207

C₂²

#2143

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

#1490

yes

characteristic

216

C₄

#1496

yes

characteristic

217

C₄

#1955

yes

characteristic

218

C₄

#1980

yes

characteristic

219

C₄

#2208

yes

—

220

C₄

#2209

yes

—

221

C₂

#9678

yes

characteristic

222

C₂

#11658

yes

characteristic

223

C₂

#11662

yes

characteristic

224

C₂

#11672

yes

characteristic

225

C₂

#11676

yes

characteristic

226

C₂

#13251

yes

characteristic

227

C₂

#13253

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×#102

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#494	2
C₄×C₄⋊C₁₆	2
C₂×#1332	2
#8274	2

subgroup

index

C₄×#494

C₄×C₄⋊C₁₆

C₂×#1332

#8274

Subgroups — #140865

Conjugacy classes of subgroups

Maximal subgroups